In 2013/14, we implemented a two-phase experimental design in a P. zonale breeding program to assess production related traits of v = 500 genotypes. In Phase 1 (P1), conducted in location 1, stock plants of genotypes were cultivated, from which the stem cutting count was obtained. In Phase 2 (P2), the stem cuttings harvested from genotypes during P1 were planted to assess the root formation in location 2. Both phases took place in greenhouses.

In P1, an α-design with r = 4 replicates, each with b = 167 incomplete blocks of size k = 3, was used. One of the incomplete blocks in each replicate was only of size two. Each experimental unit in P1 (EU1) contained a pair of stock plants from the same genotype, for a total of six plants per incomplete block of size three. Each cultivation table accommodated a full set of genotypes, i.e., one replicate.

In P2, randomization was carried out on site as follows: First, the total experimental space of rooting tables was divided into four regions. To each region in P2 the replicates of P1 were systematically assigned. A rooting table held 36 trays and a region held 72 up to 108 trays, hence, not all trays fit necessarily on one rooting table. A tray held 39 paper pots arranged in three rows. Trays were divided into areas representing EU2. Second, all stem cuttings of a genotype and replicate in P1 were packed in a small bag to transfer the plant material from location 1 to 2, and were randomly allocated to one area in the corresponding region to which a replicate was assigned in P2. Thus, stem cuttings from each EU1 were allocated exactly to one EU2. The sizes of areas for EU2 varied depending on the harvested stem cutting count per genotypes. The rooting tables and trays in P2 were considered as post-blocking factors in the analysis, which could be regarded as incomplete blocks in an IBD, for which the design was previously not optimized (Figure 1).

We idealized the experimental conditions in several ways to different degrees in the designs to be described in this and the following Sub-section “Different Block Structures in Both Phases” for comparing different design generation scenarios. This was done in the interest of focusing on the general principles implemented in scenarios investigated in this study without having to focus on intricate specifics of the P. zonale study. First, we assumed that in each phase the same block structure can be used. Thus, for each pair of stock plants we presumed that no stock plants were lost and that the stem cutting counts were reduced to six stem cuttings per genotype in P1 to assess root formation in P2. Hence, the EU2 (areas) were assumed to be of equal size in P2.

Further, the physical unit of a tray should correspond exactly to one randomization unit in P2, i.e., to an incomplete block of size six. To consider the same block structure in both phases so that block units in P1 correspond to block units of the same size in P2, requires increasing the block size in P1 from k = 3 to k = 6. These idealized conditions enabled us to assume the same resolvable IBD design with r = 4 replicates with each b = 84 incomplete blocks having the same block sizes k = 6 in both phases (Figure 1). We also assumed equal block sizes and the genotype number was increased from v = 500 to 504. Given these design properties, two options were considered to generate the two-phase experimental design (Table 1): Option 1 was to generate a design for each phase separately (Section “Option 1 – Design Generation for Each Phase Separately”) and Option 2 was to generate a design across phases by simultaneously accounting for the block structure of both phases (Section “Option 2 – Design Generation across the Two Phases”). During design generation using either Option 1 or Option 2, the complete replicates from P1 were kept intact in P2, except for Scenario III.

The assumption of a common block structure in both phases is rather idealized as in practice the phases of the experiments take place in totally different locations and different environmental conditions prevail in each phase (Molenaar et al., 2017). To account better for the environmental conditions in location 1, two different post-blocking factors were considered in the former analysis, which are now further considered in generating designs.

Designs generated by the procedure OPTEX are optimized for D-efficiency (OD or DiGGer packages provide algorithms for generating A-optimal designs). In plant breeding A-optimal or A-efficient designs are preferred as optimizing this criterion minimizes the average variance of genotype differences (MVD) (Hinkelmann and Kempthorne, 2005). Thus, the precision of estimates of genotype differences is increased and better phenotypic selection and varietal improvement can be achieved. Both D- and A-optimality usually lead to similar designs for comparative experiments with a single treatment factor (John and Williams, 1995), so the procedure OPTEX was a useful tool for our purposes, despite its focus on D-optimality. Thus, we computed the MVD obtained by linear mixed models either by intra-block or joint inter-block-intra-block dummy analyses (Piepho, 2015), in which the information about the precision of genotype parameters is contained in the variance-covariance matrix (Mead, 1988). Generally, the design showing the lowest MVD was preferred.

In all scenarios resolvability was achieved by setting the prior value for incomplete block effects in the block model specification to the total number of experimental units, 2016, except for Scenarios V, X, XII a–c and XIV a–c. Resolvability was realized for those scenarios by increasing the prior value to 10⁶.

The largest VC was the residual error variance with a proportion of 53.44% of the total variance, followed by the replicate effect with 37.56 % (Table 4). By comparison, the variance of the incomplete block effect in P1 was small (1.8%). After adding the column post-blocking factor to the first phase and still using a resolvable IBD in P2 [model (5)], the residual error variance could be reduced from 53.5 to 52.8%. The proportion of variation explained by the rows was below 1%, whereas the proportion of variation explained by columns was 2.8%. The residual error variance was further reduced by accounting for the block factor “worker-day” in P1 to 50.8%, where simultaneously the proportion of variation explained by row and column effects was reduced to below 1% [model (6) a–c]. Subsequently, scenarios were considered, where the “worker-day” was the only block factor in P1 and the resolvable IBD was retained in P2 [model (7) a–c]. By doing this, the proportion of variation explained by the replicate effects was retained, the proportion of variation explained by the “worker-day” effect in P1, and the proportion of variation explained by the incomplete block effect in P2, were maximized. However, the proportion of variation captured by the residual error was increased again by 1% compared to model (6) a–c.

In post hoc analysis of the previous experiment, the greatest MVD_(F) and MVD_(R) were observed generally in all conducted dummy analyses for model (4) (Table 5). By adding a column factor in the first phase [model (5)], a reduction of MVD_(F) was achieved by more than 50%, whereas the reduction in MVD_(R) was rather small. By considering the “worker-day” factor [model (6) a–c] in addition to the column factor, the reduction in MVD was about 0.04. When only the “worker-day” factor in the first phase [models (7), a–c] was considered, the reduction in MVD_(F) was below 3.0, whereas the MVD_(R) was slightly higher than the MVD_(R) obtained by models (6), a–c (Table 5).

For the interpretation of the reduction in MVD when comparing the alternative approaches with the two-phase design in 2013/14, the idealized conditions need to be acknowledged. The reduction in the number of incomplete blocks leads to an increased number of direct genotype comparisons within incomplete blocks which also reduces the MVD.

As expected, the MVD_(R) was always smaller than the MVD_(F) as the estimation of MVD_(R) is based not only on the within-block genotype differences (i.e., intra-block information) to obtain adjusted means like for the MVD_(F), but also on the information of block sums (i.e., the inter-block information). This stresses the importance of considering the joint inter-block–intra-block analysis (Mead et al., 2012), which can be implemented by taking blocks as random, during the design evaluation (Möhring et al., 2015). Note that, when instead of the VC of the former experiment, very large values are used for the variance of block effects, while leaving the value of residual error variance unchanged during the dummy analysis, i.e., there is no inter-block information, then values of MVD_(R) and MVD_(F) coincide. Further, the MVD_(R) varies depending on the values of VCs for block effects considered in dummy analyses, but the ranks of scenarios remained unchanged in the cases we investigated (Supplementary Presentation 4).

Randomization is conducted to avoid systematic effects and other biases in single-phase experiments (Piepho et al., 2013). In two phase experiments, these problems exist in both phases and therefore randomization should be carried out in both phases. In Scenario I, we omitted randomization in the second phase and increased thereby the efficiency of analysis, because only one incomplete block adjustment was needed to estimate the genotypes effects across the two phases [compare Model (2) and Model (4)]. That is why Scenario I showed the smallest MVD compared to the other designs assuming in each phase the same block structure. In conclusion, it is actually advantageous to transmit the experimental layout from one phase to the other whenever possible.

Two-phase designs should be generated across phases (Option 2) rather than in phase-wise order (Option 1) to guarantee the smallest MVD, which was most frequently obtained for Option 2. The only exception was Scenario I. A reason for the better performance of Option 2 is that the block structure in both phases is taken into account simultaneously when sets of genotypes are allocated to them. Thus, treatment concurrences occur equally often or only once across phases in an optimized two-phase design (Supplementary Presentation 4), which is known to be optimal in single phase experiment (John and Williams, 1995). Further, we demonstrated in the Scenarios XIII to XIV under Option 2 that our proposed approaches for generating two-phase designs across phases can be adjusted to any block size in each phase if necessary, making our approaches relevant to a broad range of applications. For the generation of a two-phase design with eight replicates (Scenario III) a similar MVD_(R) or even a smaller MVD_(F) was found than for the approach using a phase specific dummy variable (Scenario IV). But Scenario III was not further considered, as this approach restricts block structures to be the same in both phases. However, Scenario III represents an option for estimating of VCs for design effects in each phase is of interest when the structure in both phases is the same.

Examples for the application of our approaches in other ornamental species are the evaluation of rooting in P1 and other phenotypic traits in P2 in Osteospermum or the evaluation of germination rate and flowering time in the first and second phase in Dianthus ssp. (Selecta one). In the former example, in each phase the same block structure was considered, whereas in the latter example, in each phase a different block was assumed.

The greatest reduction in MVD was obtained when we accounted for the worker-induced variation by blocks in post hoc analysis of the previous experiment by models (6), a–c to (7), a–c (Table 5), although the reduction of error variance was relatively small (Table 4). This shows that workers are a source of variation and reaffirms the recommendation that known sources of variation should be captured by blocking and considered before the experiment is conducted, as precision of genotype comparisons will be increased (Mead et al., 2012).

The notable reduction in MVD_(F) and MVD_(R) realized by the alternative approaches justifies the implementation of idealized conditions in the P. zonale breeding program, especially the use of a pre-defined layout in the second phase. Under these idealized conditions, the breeder needs to randomly select six out of the total of harvested stem cuttings per pair of stock plants and genotypes in P1, which shall be rooted in P2. The procedure of packaging genotypes remains essentially the same as in the previous experiment, where the harvested stem cuttings of each genotype and replicate are packed in small bags such that each bag contained the six randomly selected stem cuttings from the EU1 in the first phase. However, the bags are now ordered according to the planting positions in P2 and then packed into cartons, where genotypes are grouped by replicate. In P2, an efficient workflow is ensured and hence, plant quality is maintained as workers plant genotypes onto trays according the planting number.